The triple cross product in R7

In this paper we consider R7 equipped with a cross product and derive an interesting formula for the triple cross product of 7-vectors, which in turn has many applications and plays an important role througout the whole paper. The relationship between G2 and the Cayley numbers was brought to the attention of physicists several years ago by Günaydin and Gürsey [2], in the context of quark structures. Here we also exploit the relationship between G2 and R7 equipped with a cross product.